March  2020, 19(3): 1421-1448. doi: 10.3934/cpaa.2020052

Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data

1. 

School of Mathematics, South China University of Technology, Guangzhou 510641, China

2. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

* Corresponding author

Received  April 2019 Revised  June 2019 Published  November 2019

In this paper, we study the asymptotic behavior of global spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations for the viscous heat conducting ideal polytropic gas flow with large initial data in $ H^1 $, when the heat conductivity coefficient depends on the temperature, practically, $ \kappa(\theta) = \tilde{\kappa}_1+\tilde{\kappa}_2\theta^q $ where constants $ \tilde{\kappa}_1>0 $, $ \tilde{\kappa}_2>0 $ and $ q>0 $ (as to the case of $ \tilde{\kappa}_1 = 0 $, please refer to the Appendix). In addition, the exponential decay rate of solutions toward to the constant state as time tends to infinity for the initial boundary value problem in bounded domain is obtained. The mass density and temperature are proved to be pointwise bounded from below and above, independent of time although strong nonlinearity in heat diffusion. The analysis is based on some delicate uniform energy estimates independent of time.

Citation: Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052
References:
[1] S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, Cambridge, 1990. 
[2]

H. B. Cui and Z. A. Yao, Asympyopic behavior of compressible p-th power Newtonian fluid with large initial data, J. Differential Equations, 258 (2015), 919-953.  doi: 10.1016/j.jde.2014.10.011.

[3]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408.  doi: 10.1137/0513029.

[4]

R. DuanA. Guo and C. J. Zhu, Global strong solution to compressible Navier-Stokes equations with density dependent viscosity and temperature dependent heat conductivity, J. Differential Equations, 262 (2017), 4314-4335.  doi: 10.1016/j.jde.2017.01.007.

[5]

H. Frid and V. Shelukhin, Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry, SIAM J. Math. Anal., 31 (2000), 1144-1156.  doi: 10.1137/S003614109834394X.

[6]

D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Ration. Mech. Anal., 173 (2004), 297-343.  doi: 10.1007/s00205-004-0318-5.

[7]

L. Hsiao and T. Luo, Large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 1277-1296.  doi: 10.1017/S0308210500023404.

[8]

H. K. Jenssen and T. K. Karper, One-dimensional compressible flow with temperature dependent transport coefficients, SIAM J. Math. Anal., 42 (2010), 904-930.  doi: 10.1137/090763135.

[9]

S. Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z., 216 (1994), 317-336.  doi: 10.1007/BF02572324.

[10]

S. Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas, Ann. Mat. Pura Appl., 175 (1998), 253-275.  doi: 10.1007/BF01783686.

[11]

S. Jiang, Global smooth solutions of the equations of a viscous, heat-conducting, one-dimensional gas with density-dependent viscosity, Math. Nachr., 190 (1998), 169-183.  doi: 10.1002/mana.19981900109.

[12]

S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193.  doi: 10.1007/s002200050526.

[13]

S. Jiang and J. W. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal., 41 (2009), 237-268.  doi: 10.1137/07070005X.

[14]

B. Kawohl, Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.

[15]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.

[16]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2$^{nd}$ edition, Pergamon Press Ltd., OXford, 1987.

[17]

Z. L. Li and Z. H. Guo, On free boundary problem for compressible Navier-Stokes equations with temperature-dependent heat conductivity, Discrete Contin.Dyn.Syst.-B, 22 (2017), 3903-3919.  doi: 10.3934/dcdsb.2017201.

[18]

J. Li and Z. L. Liang, Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier-Stokes system in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208.  doi: 10.1007/s00205-015-0952-0.

[19]

H.X. LiuT. YangH. J. Zhao and Q. Y. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data, SIAM J. Math. Anal., 46 (2014), 2185-2228.  doi: 10.1137/130920617.

[20]

T. Nagasawa, Global asymptotics of the outer pressure problem with free boundary, Japan J. Appl. Math., 5 (1988), 205-224.  doi: 10.1007/BF03167873.

[21]

V. B. Nikolaev, Solvability of a mixed problem for equations of one-dimensional axisymmetric motion of a viscous gas, Dinamika Sploshn. Sredy, 175 (1980), 83-92. 

[22]

R. H. Pan and W. Z. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.  doi: 10.4310/CMS.2015.v13.n2.a7.

[23]

X. L. Qin and Z. A. Yao, Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 244 (2008), 2041-2061.  doi: 10.1016/j.jde.2007.11.001.

[24]

X. L. QinT. YangZ. A. Yao and W. S. Zhou, Vanishing shear viscosity and boundary layer for the Navier-Stokes equations with cylindrical symmetry, Arch. Ration. Mech. Anal., 216 (2015), 1049-1086.  doi: 10.1007/s00205-014-0826-x.

[25]

Y. M. Qin, Exponential stability for the compressible Navier-Stokes equations with the cylinder symmetry in $R^3$, Nonlinear Anal. Real World Appl., 11 (2010), 3590-3607.  doi: 10.1016/j.nonrwa.2010.01.006.

[26]

Y. M. Qin and L. M. Jiang, Global existence and exponential stability of solutions in $H^4$ for the compressible Navier-Stokes equations with the cylinder symmetry, J. Differential Equations, 249 (2010), 1353-1384.  doi: 10.1016/j.jde.2010.05.019.

[27]

L. Wan and T. Wang, Asymptotic behavior for cylindrically symmetric nonbarotropic flows in exterior domains with large data, Nonlinear Anal. Real World Appl., 39 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.06.006.

[28]

L. Wan and T. Wang, Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients, J. Differential Equations, 262 (2017), 5939-5977.  doi: 10.1016/j.jde.2017.02.022.

[29]

T. Wang and H. J. Zhao, One-dimensional compressible heat-conducting gas with temperature-dependent viscosity, Math. Models Methods Appl. Sci., 26 (2016), 2237-2275.  doi: 10.1142/S0218202516500524.

[30] Y. B. Zekdovich and Y. P. Raizer, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, vol. II, Academic Press, New York, 1967. 

show all references

References:
[1] S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, Cambridge, 1990. 
[2]

H. B. Cui and Z. A. Yao, Asympyopic behavior of compressible p-th power Newtonian fluid with large initial data, J. Differential Equations, 258 (2015), 919-953.  doi: 10.1016/j.jde.2014.10.011.

[3]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408.  doi: 10.1137/0513029.

[4]

R. DuanA. Guo and C. J. Zhu, Global strong solution to compressible Navier-Stokes equations with density dependent viscosity and temperature dependent heat conductivity, J. Differential Equations, 262 (2017), 4314-4335.  doi: 10.1016/j.jde.2017.01.007.

[5]

H. Frid and V. Shelukhin, Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry, SIAM J. Math. Anal., 31 (2000), 1144-1156.  doi: 10.1137/S003614109834394X.

[6]

D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Ration. Mech. Anal., 173 (2004), 297-343.  doi: 10.1007/s00205-004-0318-5.

[7]

L. Hsiao and T. Luo, Large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 1277-1296.  doi: 10.1017/S0308210500023404.

[8]

H. K. Jenssen and T. K. Karper, One-dimensional compressible flow with temperature dependent transport coefficients, SIAM J. Math. Anal., 42 (2010), 904-930.  doi: 10.1137/090763135.

[9]

S. Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z., 216 (1994), 317-336.  doi: 10.1007/BF02572324.

[10]

S. Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas, Ann. Mat. Pura Appl., 175 (1998), 253-275.  doi: 10.1007/BF01783686.

[11]

S. Jiang, Global smooth solutions of the equations of a viscous, heat-conducting, one-dimensional gas with density-dependent viscosity, Math. Nachr., 190 (1998), 169-183.  doi: 10.1002/mana.19981900109.

[12]

S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193.  doi: 10.1007/s002200050526.

[13]

S. Jiang and J. W. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal., 41 (2009), 237-268.  doi: 10.1137/07070005X.

[14]

B. Kawohl, Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.

[15]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.

[16]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2$^{nd}$ edition, Pergamon Press Ltd., OXford, 1987.

[17]

Z. L. Li and Z. H. Guo, On free boundary problem for compressible Navier-Stokes equations with temperature-dependent heat conductivity, Discrete Contin.Dyn.Syst.-B, 22 (2017), 3903-3919.  doi: 10.3934/dcdsb.2017201.

[18]

J. Li and Z. L. Liang, Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier-Stokes system in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208.  doi: 10.1007/s00205-015-0952-0.

[19]

H.X. LiuT. YangH. J. Zhao and Q. Y. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data, SIAM J. Math. Anal., 46 (2014), 2185-2228.  doi: 10.1137/130920617.

[20]

T. Nagasawa, Global asymptotics of the outer pressure problem with free boundary, Japan J. Appl. Math., 5 (1988), 205-224.  doi: 10.1007/BF03167873.

[21]

V. B. Nikolaev, Solvability of a mixed problem for equations of one-dimensional axisymmetric motion of a viscous gas, Dinamika Sploshn. Sredy, 175 (1980), 83-92. 

[22]

R. H. Pan and W. Z. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.  doi: 10.4310/CMS.2015.v13.n2.a7.

[23]

X. L. Qin and Z. A. Yao, Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 244 (2008), 2041-2061.  doi: 10.1016/j.jde.2007.11.001.

[24]

X. L. QinT. YangZ. A. Yao and W. S. Zhou, Vanishing shear viscosity and boundary layer for the Navier-Stokes equations with cylindrical symmetry, Arch. Ration. Mech. Anal., 216 (2015), 1049-1086.  doi: 10.1007/s00205-014-0826-x.

[25]

Y. M. Qin, Exponential stability for the compressible Navier-Stokes equations with the cylinder symmetry in $R^3$, Nonlinear Anal. Real World Appl., 11 (2010), 3590-3607.  doi: 10.1016/j.nonrwa.2010.01.006.

[26]

Y. M. Qin and L. M. Jiang, Global existence and exponential stability of solutions in $H^4$ for the compressible Navier-Stokes equations with the cylinder symmetry, J. Differential Equations, 249 (2010), 1353-1384.  doi: 10.1016/j.jde.2010.05.019.

[27]

L. Wan and T. Wang, Asymptotic behavior for cylindrically symmetric nonbarotropic flows in exterior domains with large data, Nonlinear Anal. Real World Appl., 39 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.06.006.

[28]

L. Wan and T. Wang, Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients, J. Differential Equations, 262 (2017), 5939-5977.  doi: 10.1016/j.jde.2017.02.022.

[29]

T. Wang and H. J. Zhao, One-dimensional compressible heat-conducting gas with temperature-dependent viscosity, Math. Models Methods Appl. Sci., 26 (2016), 2237-2275.  doi: 10.1142/S0218202516500524.

[30] Y. B. Zekdovich and Y. P. Raizer, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, vol. II, Academic Press, New York, 1967. 
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